I would say that I don't think that equation would hold, in general, at all. On the LHS, you've got 's and 's left over after the integrations. On the RHS, you've only got 's left over after the integrations. A better question might be, does the equation

hold, in general? Are you sure you didn't mean to ask this? Because this equation has a chance of being correct.

Let me ask this question: from where did you get these equations that you're trying to disprove? Are they a theorem from somewhere? If so, where?

Yeah exactly,I doubt at the truth of the second equation you wrote(the one with "dy dx" in the last
I have read the same equation in the book "INTEGRAL EQUATIONS- A SHORT COURSE" by G.CHAMBERS in the chapter "VOLTERRA INTEGRAL EQUATIONS" under the topic named "CONVOLUTION TYPE KERNELS"